Hi,
I am an international student in the UK. I will be finishing my last BA year in a few months. Is it easy for international students at times to get offers for graduate course in the top unis: Oxbridge, LSE, UCL or Warwick, because they pay more at some courses? or they do not care at ll. because I heard this from my tutor in current average uni that such unis will offer u a place because you are paying them for example 27,000 pounds in Msc Economics at LSE.
So it really true or they do not care at all.

Well, put it this way: it's not going to negatively affect your application. No university would admit to accepting international students over domestic students in such a manner to get more money, but it may happen.

More likely, international students who have the financial resources to do an undergraduate course, probably also have the resources to pay for a masters course. UK students however normally rely on the government funded loans scheme, and while this has been extended to masters courses in recent years, doesn't cover the full amount a student is likely to need to complete such a course, so necessarily fewer domestic students will be applying to such graduate courses (as compared to undergrad), opposed to a (relatively) static number of international applicants for both sets - which accounts for the disparity in international vs domestic student numbers in such courses. Additionally those domestic students who have a great deal of financial reserves may be more inclined to do their masters in another country (such as the US or in the EU).

I doubt it will have much affect on an application either way, although you may well find there are more international students, proportionally, in your cohort than as an undergrad.

Certainly for Oxbridge (and likely for other unis as well), it's no easier for an international student to get an offer than anyone else. Everyone goes through the same process of personal statements, admissions tests and interviews (or the equivalent process for postgrads), and offers are made based solely on those and on your academic performance. Small, less well known universities might well be attracted to the increased revenue international students bring, though.